# Properties

 Label 405600bf Number of curves $2$ Conductor $405600$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bf1")

sage: E.isogeny_class()

## Elliptic curves in class 405600bf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
405600.bf2 405600bf1 $$[0, -1, 0, 40842, -908688]$$ $$1560896/975$$ $$-4706138775000000$$ $$$$ $$1548288$$ $$1.6964$$ $$\Gamma_0(N)$$-optimal*
405600.bf1 405600bf2 $$[0, -1, 0, -170408, -7246188]$$ $$14172488/7605$$ $$293663059560000000$$ $$$$ $$3096576$$ $$2.0430$$ $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 2 curves highlighted, and conditionally curve 405600bf1.

## Rank

sage: E.rank()

The elliptic curves in class 405600bf have rank $$1$$.

## Complex multiplication

The elliptic curves in class 405600bf do not have complex multiplication.

## Modular form 405600.2.a.bf

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{7} + q^{9} - 4q^{17} + 2q^{19} + O(q^{20})$$ ## Isogeny matrix

sage: E.isogeny_class().matrix()

The $$i,j$$ entry is the smallest degree of a cyclic isogeny between the $$i$$-th and $$j$$-th curve in the isogeny class, in the Cremona numbering.

$$\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)$$

## Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)

The vertices are labelled with Cremona labels. 